Matter waves in a gravitational field: An index of refraction for massive particles in general relativity

American Journal of Physics

Volumn 69, Issue 10, 2001, Pages 1103-1110

  Evans, James ^a   Alsing, Paul M ^b   Giorgetti, Stefano ^c   Nandi, Kamal Kanti ^d  

^a UNIVERSITY OF PUGET SOUND   (United States)

^b UNIVERSITY OF NEW MEXICO   (United States)

^c UNIVERSITY OF MILAN   (Italy)

^d UNIVERSITY OF NORTH BENGAL   (India)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035621002     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.1389281     Document Type: Article

References (26)

1. For a systematic technique for finding isotropic coordinates, see R. Adler, M. Bazin, and M. Schiffer, Introduction to General Relativity and Gravitation (McGraw-Hill, New York, 1965), pp. 174-177. Many examples of metrics that can be put into isotropic form are given in Ref. 5.
2. The idea of representing the gravitational field as an optical medium with an effective index of refraction goes all the way back to the early days of general relativity. A. S. Eddington, Space, Time and Gravitation (Cambridge U.P., Cambridge, 1920), p. 109.
3. This optical theory has a long history, going back to Bernoulli and Maxwell. For a systematic development see J. Evans and M. Rosenquist, " 'F = ma' optics," Am. J. Phys. 54, 876-883 (1986); J. Evans, "Simple forms for equations of rays in gradient-index lenses," ibid. 58, 773-778 (1990).
4. This optical theory has a long history, going back to Bernoulli and Maxwell. For a systematic development see J. Evans and M. Rosenquist, " 'F = ma' optics," Am. J. Phys. 54, 876-883 (1986); J. Evans, "Simple forms for equations of rays in gradient-index lenses," ibid. 58, 773-778 (1990).
5. K. K. Nandi and A. Islam, "On the optical-mechanical analogy in general relativity," Am. J. Phys. 63, 251-256 (1995).
6. This section briefly summarizes results given in J. Evans, K. K. Nandi, and A. Islam, "On the optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons," Gen. Relativ. Gravit. 28, 413-439 (1996).
7. For a simpler introduction, with examples based on the Schwarzschild metric, see J. Evans, K. K. Nandi, and A. Islam, "The optical-mechanical analogy in general relativity: New methods for the paths of light and of the planets," Am. J. Phys. 64, 1404-1415 (1996).
8. For an extension of the theory to a broader class of metrics see P. M. Alsing, "The optical-mechanical analogy for stationary metrics in general relativity," Am. J. Phys. 66, 779-790 (1998).
9. Louis de Broglie, Recherches sur la Théorie des Quanta (Masson, Paris, 1963), a reprint of the thesis of 1924.
10. See Ref. 5, pp. 435-437.
11. For problems concerning quantization in curved space see the discussions in N. Birrell and P. Davies, Quantum Fields in Curved Space (Cambridge U. P., New York, 1982) and in S. A. Fulling, "Nonuniqueness of canonical field quantization in a Riemannian space-time," Phys. Rev. D 7, 2850-2862 (1973); D. W. Sciama, P. Candellas, and D. Deutsch, "Quantum field theory, horizons and thermodynamics," Adv. Phys. 30, 327-366 (1981); S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time (Cambridge U.P., New York, 1989).
12. For problems concerning quantization in curved space see the discussions in N. Birrell and P. Davies, Quantum Fields in Curved Space (Cambridge U. P., New York, 1982) and in S. A. Fulling, "Nonuniqueness of canonical field quantization in a Riemannian space-time," Phys. Rev. D 7, 2850-2862 (1973); D. W. Sciama, P. Candellas, and D. Deutsch, "Quantum field theory, horizons and thermodynamics," Adv. Phys. 30, 327-366 (1981); S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time (Cambridge U.P., New York, 1989).
13. For problems concerning quantization in curved space see the discussions in N. Birrell and P. Davies, Quantum Fields in Curved Space (Cambridge U. P., New York, 1982) and in S. A. Fulling, "Nonuniqueness of canonical field quantization in a Riemannian space-time," Phys. Rev. D 7, 2850-2862 (1973); D. W. Sciama, P. Candellas, and D. Deutsch, "Quantum field theory, horizons and thermodynamics," Adv. Phys. 30, 327-366 (1981); S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time (Cambridge U.P., New York, 1989).
14. For problems concerning quantization in curved space see the discussions in N. Birrell and P. Davies, Quantum Fields in Curved Space (Cambridge U. P., New York, 1982) and in S. A. Fulling, "Nonuniqueness of canonical field quantization in a Riemannian space-time," Phys. Rev. D 7, 2850-2862 (1973); D. W. Sciama, P. Candellas, and D. Deutsch, "Quantum field theory, horizons and thermodynamics," Adv. Phys. 30, 327-366 (1981); S. A. Fulling, Aspects of Quantum Field Theory in Curved Space-Time (Cambridge U.P., New York, 1989).
15. A solution for the wave function in the limit of weak gravitational fields is given by J. Donoghue and B. Holstein, "Quantum mechanics in curved space," Am. J. Phys. 54, 827-831 (1986).
16. L. Landau and E. Lifshitz, The Classical Theory of Fields (Addison-Wesley, Reading, MA, 1965), p. 244.
17. For example, in Ref. 11.
18. M. H. Holmes, Introduction to Perturbation Methods (Springer-Verlag, New York, 1995), pp. 197-207.
19. M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1990), 6th ed., pp. 114-115.
20. For an introduction to this subject, see D. Greenberger and A. W. Overhauser, "Coherent effects in neutron diffraction and gravity experiments," Rev. Mod. Phys. 51, 43-78 (1979) and L. Stodolsky, "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979).
21. For an introduction to this subject, see D. Greenberger and A. W. Overhauser, "Coherent effects in neutron diffraction and gravity experiments," Rev. Mod. Phys. 51, 43-78 (1979) and L. Stodolsky, "Matter and light wave interferometry in gravitational fields," Gen. Relativ. Gravit. 11, 391-405 (1979).
22. N. Fornengo, C. Giunti, C. W. Kirn, and J. Song, "Gravitational effects on the neutrino oscillations," Phys. Rev. D 56, 1895-1902 (1997); T. Bhattacharya, S. Habib, and E. Mottola, "Gravitationally Induced Neutrino-Oscillation Phases in Static Spacetimes," ibid. 59, 067301, 4 pages (1999).
23. N. Fornengo, C. Giunti, C. W. Kirn, and J. Song, "Gravitational effects on the neutrino oscillations," Phys. Rev. D 56, 1895-1902 (1997); T. Bhattacharya, S. Habib, and E. Mottola, "Gravitationally Induced Neutrino-Oscillation Phases in Static Spacetimes," ibid. 59, 067301, 4 pages (1999).
24. P. M. Alsing, J. C. Evans, and K. K. Nandi, "The phase of a quantum mechanical particle in curved spacetime," Gen. Relativ. Gravit. (in press). Preprint at http://arXiv.org, gr-qc0010065 (2000).
25. Reference 15, Eq. (38), p. 117.
26. R. D'Inverno, Introducing Einstein's Relativity (Clarendon, New York, 1992), pp. 93-94, see also Ref. 12, pp. 242-243.